Would a 16-bit image for a bump map have any advantage?
8 bit images normally store 2^8=256 different shades of grey. In theory, you could encode more information in it, but most displacement textures you use wont. This means, that if your displaced model is 8 Blender units high, the smallest height difference you will get is 8/256=1/8th Blender unit. This doesn't seem like much, but can create visible steps between two heights.

Using a 16 bit image can store 2^16=65536 different height levels. No matter how you use your bump texture, you won't notice any artifacts here.